Assessment and Future Directions of Nonlinear Model Predictive ...

MPC for Stochastic Systems 259we haveE k l(k + j|k +1)=l(k + j|k) − (κ 2 1 − 1)( σ 2 1 (k + j|k) − σ2 1 (k + j|k +1)) .The required bound therefore holds if κ 1 ≥ 1 since σ 2 1(k + j|k) ≥ σ 2 1(k + j|k +1).Remark 1. In accordance with Lemma 1 it is assumed below that κ 1 ≥ 1, orequivalently that the bounds (6) are invoked with probability p 1 ≥ 84.1% (to 3s.f.). With κ 1 = 1, this formulation recovers the conventional expectation cost:l(k + j|k) =E k y1 2 (k + j|k) for regulation problems.Consider next the definition of constraints. Since output predictions are Gaussianrandom variables, we consider probabilistic (as opposed to hard) constraints:Pr ( y 2 (k + j|k) ≤ Y 2)≥ p2 (8)where Y 2 is a constraint threshold. Input constraints are assumed to have theform:|u(k + j|k)| ≤U (9)where u(k + j|k) is the predicted value of u(k + j) at time k.4 Terminal Constraint SetFollowing the conventional dual mode prediction paradigm [14], predicted inputtrajectories are switched to a linear terminal control law: u(k + j|k) =Kx(k + j|k), j ≥ N after an initial N-step prediction horizon. For the caseof uncertainty in the output map (2), an ellipsoidal terminal constraint canbe computed by formulating conditions for invariance and satisfaction of constraints(8),(9) under the terminal control law as LMIs [12]. However, in the caseof the model (3), the uncertainty in the predicted state trajectory requires thata probabilistic invariance property is used in place of the usual deterministicdefinition of invariance when defining a terminal constraint set. We thereforeimpose the terminal constraint that x(k + N|k) lie in a terminal set Ω witha given probability, where Ω is designed so that the probability of remainingwithin Ω under the closed-loop dynamics x(k +1)=Φ(k)x(k) isatleastp Ω , i.e.Pr(Φx ∈ Ω) ≥ p Ω ∀x ∈ Ω. (10)If constraints on the input and secondary output are satisfied everywhere withinΩ, then this approach can be used to define a receding horizon optimizationwhich is feasible with a specified probability at time k + 1 if it is feasible at timek. Given that the uncertain parameters of (3) are not assumed bounded, this isarguably the strongest form of recursive feasibility attainable.For computational convenience we consider polytopic terminal sets defined byΩ = {x : vi T x ≤ 1, i=1,...,m}. Denote the closed-loop dynamics of (3) underu = Kx as